A transition matrix for a three-state homogeneous Markov chain is given by
A process starts in state 0. Annuity contract 1 provides for periodic payments, provided the process is in state 1. Annuity contract 2 provides for periodic payments which starts at the end of the period in which process enters state 2, but stop completely upon exit from state 2 and do not begin again, even upon subsequent return. For each contract, find the probability that a payment is made at time 3.